select the graph of y = 2 sin x + 2.

Answer:

A.

Explanation:

Step1: Analyze the general form of sine - function

The general form of a sine - function is $y = A\sin(Bx - C)+D$. For the function $y = 2\sin x+2$, we have $A = 2$, $B = 1$, $C = 0$, and $D = 2$.

Step2: Determine the amplitude

The amplitude of the function $y = A\sin(Bx - C)+D$ is given by $|A|$. Here, $|A|=|2| = 2$. This means the graph of $y=\sin x$ is vertically stretched by a factor of 2.

Step3: Determine the vertical shift

The value of $D$ in the general form represents the vertical shift. Since $D = 2$, the graph of $y = 2\sin x$ is shifted 2 units up.

Step4: Evaluate at key - points

When $x = 0$, $y=2\sin(0)+2=2$. When $x=\frac{\pi}{2}$, $y = 2\sin(\frac{\pi}{2})+2=2\times1 + 2=4$. When $x=\pi$, $y=2\sin(\pi)+2=2$. When $x=\frac{3\pi}{2}$, $y=2\sin(\frac{3\pi}{2})+2=2\times(- 1)+2 = 0$. When $x = 2\pi$, $y=2\sin(2\pi)+2=2$. Graph A matches these characteristics.